Robust and Scalable Models of Microbiome Dynamics for - - PowerPoint PPT Presentation

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Robust and Scalable Models of Microbiome Dynamics for Bacteriotherapy Design

Travis E. Gibson1 Georg K. Gerber1,2

1Massachusetts Host Microbiome Center

Brigham and Women’s Hospital and Harvard Medical School

2Health Sciences and Technology Division Harvard-MIT

December 9, 2017 NIPS 2017 Workshop on Machine Learning in Computational Biology

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SLIDE 2

Outline

1 Background on the Human Microbiome 2 From Experimental Design to Bacteriotherapies 3 Model of microbial dynamics 4 Inference Model 5 Applications

Gerber Lab is looking for Post-docs and PhD students

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SLIDE 3

The Microbiome

1 The microbiome is the aggregate of

microorganisms that resides on or within any of a number of human tissues and biofluids:

• skin, mammary glands, placenta, seminal

fluid, uterus, ovarian follicles, lung, saliva, oral mucosa, conjunctiva, biliary and gastrointestinal tracts) [wikipedia]

2 1014 Microbes in/on your body [Sender et al.

PLoS Biology 2016]

3 3.3 million genes compared to 23,000 human

genes [Qin et al. Nature 2010]

4 Large component of the immune system 5 Play a role in a variety of human diseases:

• infections, arthritis, food allergy, cancer,

inflammatory bowel disease, neurological diseases, and obesity/diabetes

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SLIDE 4

Bacteriotherapy

Bacteriotherapy: communities of bacteria administered to patients for specific therapeutic applications

• “bugs-as-drugs”

Clostridium difficile infection

• Causes serious diarrhea (14K deaths/yr)
• Antibiotics disrupt helpful bacteria in gut
• Increasingly difficult to treat with conventional therapies (more antibiotics): 20-30%

recurrence rate Pharmacology meets Ecology

• C. diff

microbial interaction network positive microbe A produces a small molecule (metabolite) that microbe B needs negative two microbes competing for the same niche what if there were 300 bugs in the network?

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SLIDE 5

Workflow in our lab

batch experiments chemostat animal experiments

• 16S rRNA on MiSeq

(reads) for relative abundances of species

• 16S rRNA qPCR

(universal primers) for bacterial biomass time abundance

• measurements - irregular,

sparse & noisy

Interaction Network

• 300 species
• 90,000 interactions

Interaction Modules

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SLIDE 6

Microbial Dynamics

• Abundance of microbe i at time t : xt,i

dxt,i dt = αixt,i + βiix2

t,i +

• j=i

βijxt,ixt,j + dwt,i dt growth, carrying capacity, interaction, stochastic disturbance

• Convert to discrete time

xk+1,i = xk,i +

• αixk,i + βiix2

k,i +

• j=i

βijxk,ixk,j

• ∆k + (wk+1,i − wk,i)
• ∆t

discrete time step size Next we discuss the three main ingredients to our model

1 Clustering (interaction modules) 2 Edge selection (structure learning, variable selection) 3 Introduction of an auxiliary variable between the measurement model

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SLIDE 7

Complete Model

Dirichlet Process πc | α ∼ Stick(α) ci | πc ∼ Multinomial(πc) bci,cj | σb ∼ Normal(0, σ2

b)

Edge Selection (Structure) zci,cj | πz ∼ Bernouli(πz) Self Interactions ai,1, ai,2 | σa ∼ Normal(0, σ2

a)

Dynamics xk+1,i | xk, ai, b, c, z, σw ∼ Normal

• xk,i+xk,i
• ai,1+ai,2xk,i+

cj=ci

bci,cjzci,cjxk,j

• , ∆kσ2

w

• Constraint and Measurement Model

qk,i | xk,i ∼ Normal(xk,i, σ2

q)

yk,i | σy, qk,i ∼ f(qk,i) f ∈ {Neg. Bin., Log Norm., ...}

xk,i qk,i yk,i bℓ,m σb zℓ,m πz ai k ∈ [m] i ∈ [n] ci πc α σa ℓ ∈ Z+ m ∈ Z+ i ∈ [n]

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SLIDE 8

Simple example without the intermediate auxiliary variable

xt+1,i | xt, a ∼ Normal≥0(aiTf(xt), σ2

xi)

yt,i | xt,i ∼ Normal≥0(xt,i, σ2

yi)

ai ∼ Normal(0, σ2

ai) x1 x2 x3 · · · xn a Σa y1 y2 y3 · · · yn

Note the truncated dis- tributions for x and y Parameter inference Gibbs step: a(g+1) ∼ pa|x(· | x(g)) pa|x ∝ px|apx|a

Normal≥0(x; µ(a, x), σ2)

papa

Normal(a; 0, σ2)

= e−

1 2σ2 (x−µ(a,x))2

σ √ 2π

• Φ(∞) − Φ
• −µ(a, x)

σ e−

1 2σ2 a2

σ √ 2π Sampling for other variables

• Filtering (sampling from posterior of x) is challenging
• Can not use collapsed Gibbs sampling for Dirichlet Process or Edge Selection

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SLIDE 9

Introducing an auxiliary variable

xt+1,i | xt, a ∼ Normal(aT

i f(xt), σ2 xi)

qk,i | xk,i ∼ Normal(xk,i, σ2

q)

qk,i ∼ Uniform[0, L) yk,i | σy, qk,i ∼ Normal≥0(qk,i, σ2

y )

ai ∼ Normal(0, σ2

ai)

x1 x2 x3 · · · xn a Σa q1 q2 q3 · · · qn y1 y2 y3 · · · yn

Prior on q is positive, relaxing the distribution

• n the dynamics for x

Parameter inference Gibbs step: a(g+1) ∼ pa|x(· | x(g))

• Direct sampling from the posterior now possible (Bayesian Regression!)

Sampling for other variables

• Collapsed Gibbs sampling for Dirichlet Process and Edge Selection (integrate out a)
• Filtering is still challenging but easier to design proposals than before (MH)

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SLIDE 10

Synthetic consortia of small microbial community

Marika Ziesack

Silver Lab, Harvard

• E. coli
• B. fragilis
• S. typhimurium
• B. theta
• Microbes engineered to overproduces one amino acid
• Microbes engineered to need three amino acids
• Compare inference on WT and engineered strains to

prove that engineering was performed.

Synthetic Data

200 400 600 10 20 abundance Example Trajectory time 1 2 3 4 1 2 3 4 Ground Truth

• 1

1 2 2 1 1

• 4
• 3
• 4
• 3
• 4
• 2

2 4 1/(abundance time) 1 2 3 4 1 2 3 4

Bayes Factors

Inf Inf Inf 1226 Inf Inf 23.4 Inf 0.1 0.9 0.2 2.5 0.2 3 0.3 0.2

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.6

0.1 1 0.4 0.3 2.1

0.5

• 4.1 -2.5
• 3.4
• 3.1
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time 1 2 3 4 1 2 3 4 Bayes Factors

Inf Inf Inf 22.66 Inf Inf Inf Inf Inf Inf 1.194 0.08 0.24 0.107 0.172 0.191

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.9

1

0.5 2 0.7 0.6

• 4.4 -2.5
• 3.4
• 3
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time

Learning from 2 batch experiments Learning from 4 batch experiments

200 400 600 10 20 abundance Example Trajectory time 1 2 3 4 1 2 3 4 Ground Truth

• 1

1 2 2 1 1

• 4
• 3
• 4
• 3
• 4
• 2

2 4 1/(abundance time) 1 2 3 4 1 2 3 4

Bayes Factors

Inf Inf Inf 1226 Inf Inf 23.4 Inf 0.1 0.9 0.2 2.5 0.2 3 0.3 0.2

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.6

0.1 1 0.4 0.3 2.1

0.5

• 4.1 -2.5
• 3.4
• 3.1
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time 1 2 3 4 1 2 3 4 Bayes Factors

Inf Inf Inf 22.66 Inf Inf Inf Inf Inf Inf 1.194 0.08 0.24 0.107 0.172 0.191

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.9

1

0.5 2 0.7 0.6

• 4.4 -2.5
• 3.4
• 3
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time

Learning from 2 batch experiments Learning from 4 batch experiments

200 400 600 10 20 abundance Example Trajectory time 1 2 3 4 1 2 3 4 Ground Truth

• 1

1 2 2 1 1

• 4
• 3
• 4
• 3
• 4
• 2

2 4 1/(abundance time) 1 2 3 4 1 2 3 4

Bayes Factors

Inf Inf Inf 1226 Inf Inf 23.4 Inf 0.1 0.9 0.2 2.5 0.2 3 0.3 0.2

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.6

0.1 1 0.4 0.3 2.1

0.5

• 4.1 -2.5
• 3.4
• 3.1
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time 1 2 3 4 1 2 3 4 Bayes Factors

Inf Inf Inf 22.66 Inf Inf Inf Inf Inf Inf 1.194 0.08 0.24 0.107 0.172 0.191

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.9

1

0.5 2 0.7 0.6

• 4.4 -2.5
• 3.4
• 3
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time

Learning from 2 batch experiments Learning from 4 batch experiments

200 400 600 10 20 abundance Example Trajectory time 1 2 3 4 1 2 3 4 Ground Truth

• 1

1 2 2 1 1

• 4
• 3
• 4
• 3
• 4
• 2

2 4 1/(abundance time) 1 2 3 4 1 2 3 4

Bayes Factors

Inf Inf Inf 1226 Inf Inf 23.4 Inf 0.1 0.9 0.2 2.5 0.2 3 0.3 0.2

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.6

0.1 1 0.4 0.3 2.1

0.5

• 4.1 -2.5
• 3.4
• 3.1
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time 1 2 3 4 1 2 3 4 Bayes Factors

Inf Inf Inf 22.66 Inf Inf Inf Inf Inf Inf 1.194 0.08 0.24 0.107 0.172 0.191

2 4 6 8 10 1 2 3 4 1 2 3 4

Interaction Coefficients

• 0.9

1

0.5 2 0.7 0.6

• 4.4 -2.5
• 3.4
• 3
• 4
• 2

2 4 1/(abundance time) 200 400 600 10 20

abundance Simulated Trajectories time

Learning from 2 batch experiments Learning from 4 batch experiments

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SLIDE 11

Animal experiments with Clostridium difficile infection

• Colonize mice with a defined complex of 12 bacteria (GnotoComplex), then challenge with

Clostridium difficile

Day 1 Day 28 Day 56

• c. difficile

GnotoComplex

• 5 mice (26 fecal samples taken from each, 16s and universal qPCR)

1 2 3 4 5 6 7 8 9 10 11 12 13

Microbe Co-cluster Proportions

0.2 0.4 0.6 0.8 1

Klebsiella oxytoca 13 Bacteroides fragilis 12 Roseburia hominis 11 Bacteroides vulgatus 10 Parabacteroides distasonis 9 Akkermansia muciniphila 8 Ruminococcus obeum 7 Escherichia coli 6 Clostridium scindens 5 Bacteroides ovatus 4 Clostridium ramosum 3 Proteus mirabilis 2 Clostridium difficile 1 1 2 3 4 5 6 7 8 9 10 11 12 13

Microbe Interaction Strength

• 13
• 12
• 11
• 10
• 9
• 8

Klebsiella oxytoca 13 Bacteroides fragilis 12 Roseburia hominis 11 Bacteroides vulgatus 10 Parabacteroides distasonis 9 Akkermansia muciniphila 8 Ruminococcus obeum 7 Escherichia coli 6 Clostridium scindens 5 Bacteroides ovatus 4 Clostridium ramosum 3 Proteus mirabilis 2 Clostridium difficile 1 gram/(CFU day) log10

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SLIDE 12

Conclusions

We have presented

• Fully Bayesian inference model for microbial dynamics
• Interpretability features
• Reducing the microbial interaction network complexity via extraction of modular features
• Edge Selection so as to give us confidence as to what interactions are real

Future Directions

• Apply algorithm to mice that have been administered human fecal samples (complex flora

300+ species)

• Approximate Bayesian methods for dynamical systems analysis
• Modeling host dynamics (Layered latent dynamical processes)

Funding: DARPA, NIH Acknowledge: Organizers, Gerber Lab, Bry Lab, Silver Lab email: tgibson@mit.edu

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SLIDE 13

Gerber Lab Plug

Gerber Lab is looking for post-docs and PhD students

Georg K. Gerber, MD, PhD, (ggerber@bwh.harvard.edu)

• Assistant Professor, Harvard Medical School
• Co-Director, Massachusetts Host-Microbiome Center
• Member of the Harvard-MIT Health Sciences & Technology Faculty
• Associate Pathologist, Center for Advanced Molecular Diagnostics Department of

Pathology, Brigham and Women’s Hospital

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